Lumped Isothermal Method - Detailed Discussion
When this AFT Arrow Steady Solution Method is used elevation data is ignored by the AFT Arrow Steady solver. The output will still show the elevation as entered by the user. Since elevation effects are usually not very important in gas systems, neglecting elevation changes is frequently an acceptable approximation.
Another special case of flow in pipes is that of isothermal flow. In isothermal flow, the gas static temperature remains constant. The tendency is for gas to cool as it flows along a pipe. For the temperature to remain constant, an inflow of heat is required.
When the temperature remains constant, it removes one of the unknowns from Equations 1, 2 and 5. From Equation 5, the density becomes directly proportional to pressure, and an analytical solution can be obtained if elevation changes are neglected (e.g., see Saad, 1993, pp. 264-269):
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(44) |
where the T subscript on L emphasizes isothermal. Integrating from point 0 to L,
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(45) |
Isothermal flow behavior results in a different relationship for sonic choking as compared to adiabatic flow. In fact, to truly maintain isothermal flow up to the sonic point requires an infinite amount of heat transfer. This results in the strange but mathematically correct conclusion that sonic choking occurs at a Mach number less than 1 for isothermal flow. Sonic choking will occur at 1/sqrt(gamma) for isothermal flow, where gamma is the isentropic expansion coefficient (Saad, 1993, pp. 267). Practically speaking, isothermal flow will not remain isothermal at high velocities.
Similar to the Lumped Adiabatic method, Equation 45 is solved for each pipe, using an average value of friction factor and g for each pipe. The solution of Equation 45 typically is iterative where the unknown is M2.
When the user chooses the Lumped Isothermal method, an isothermal temperature is required in the AFT Arrow Steady Solution panel. This temperature is used for all pipes.
